A generalized Lane–Emden equation with indices is discussed, which reduces to the Lane–Emden equation proper for General properties of the set of solutions of this equation are derived, and exact solutions are given. These include a singular solution without free integration constant for arbitrary and for particular relations between ν, and α. Among the two-parameter solutions nonequivalent families of solutions of the same equation are obtained.
REFERENCES
1.
W. Thomson, Collected Papers (Cambridge University Press, Cambridge, 1911), Vol. 5, p. 266.
2.
R. Emden, Gaskugeln (Teubner, Leipzig and Berlin, 1907).
3.
4.
5.
S. D.
Maharaj
, P. G. L.
Leach
, and R.
Maartens
, Gen. Relativ. Gravit.
23
, 261
(1991
).6.
7.
8.
9.
10.
I. W.
Roxburgh
and L. M
Stockman
, Mon. Not. R. Astron. Soc.
303
, 466
(1999
).11.
G.
Adomian
, R.
Rach
, and N. T.
Shawagfeh
, Found. Phys. Lett.
8
, 161
(1995
).12.
13.
14.
S. Chandrasekhar, An introduction to the study of stellar structure (Dover, New York, 1957), Chap. IV.
15.
H. T. Davis, Introduction to Nonlinear Differential and Integral Equations (Dover, New York, 1962).
16.
17.
M. H. Wrubel, “Stellar Interiors,” in Encyclopedia of Physics, edited by S. Flügge (Springer-Verlag, Berlin 1958), Vol. 51, p. 53.
18.
19.
S.
Gnutzmann
and U.
Ritschel
, Z. Phys. B: Condens. Matter
96
, 391
(1995
).20.
21.
22.
23.
24.
E. Hille, Ordinary Differential Equations in the Complex Domain (Wiley, New York, 1976), Sect. 12.4.
25.
26.
27.
K. S.
Govinder
, P. G. L.
Leach
, and S. D.
Maharaj
, Int. J. Theor. Phys.
34
, 625
(1995
).28.
29.
P. Havas and H. Goenner, Spherically symmetric space–times with constant curvature scalar (unpublished).
30.
E. T. Whittaker, Analytical Dynamics, Sec. 48 (4th ed.), New York, 1944.
31.
J. D. Logan, Invariant Variational Principles (Academic, New York, 1977).
32.
B. A.
Huberman
and J. P.
Crutchfield
, Phys. Rev. Lett.
43
, 1743
(1979
);33.
34.
35.
36.
37.
J. M.
Dixon
and J. A.
Tuszynski
, Phys. Rev. A
41
, 4166
(1990
).38.
E. Kamke, Differentialgleichungen, Lösungsmethoden und Lösungen, 7th ed. (Akad. Verlagsgesellschaft, Geest & Portig, Leipzig, 1961).
39.
40.
41.
42.
R. C.
Flagg
, C. D.
Luning
, and W. L.
Perry
, J. Comput. Phys.
38
, 396
(1980
).43.
44.
45.
46.
47.
48.
V. R.
Gavrilov
and V. N.
Melnikov
, Theor. Math. Phys.
114
, 355
(1998
).49.
50.
51.
52.
53.
54.
55.
56.
I. M. Ryshik and I. S. Gradstein, Tables of Series, Products and Integrals (Deutscher Verlag der Wissenschaften, Berlin, 1957).
57.
P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin 1954).
58.
N. N. Bogoljubov und J. A. Mitroposlki, Asymptotische Methoden in der Theorie der nichtlinearen Schwingungen (Akad. Verlagsgesellschaft, Berlin, 1965), p. 6.
59.
G. P. Flessas, in Finite dimensional integrable nonlinear dynamical systems, edited by P. G. L Leach and W. H. Steeb (World Scientific, Singapore, 1988), pp. 161–173.
This content is only available via PDF.
© 2000 American Institute of Physics.
2000
American Institute of Physics
You do not currently have access to this content.
